Free Heat Engine Efficiency Calculator
Calculate the maximum theoretical (Carnot) efficiency of a heat engine from its hot and cold reservoir temperatures in kelvin, using η = 1 − Tc/Th.
Enter reservoir temperatures in kelvin to find the Carnot (maximum) efficiency.
η = 1 − Tc/Th. Use absolute temperatures: add 273.15 to a Celsius value to get kelvin. This is the maximum possible efficiency; real engines achieve less.
Quick answer
The maximum efficiency of any heat engine is the Carnot efficiency, η = 1 − Tc/Th, where Tc and Th are the cold and hot reservoir temperatures measured in kelvin. For example, an engine running between a hot reservoir at Th = 600 K and a cold reservoir at Tc = 300 K has η = 1 − 300/600 = 0.50, or 50%. No real engine operating between those two temperatures can ever exceed that limit.
Formula & method
Carnot efficiency
η = 1 − Tc / Th
- η — Maximum (Carnot) thermal efficiency, a fraction from 0 to 1
- Th — Absolute temperature of the hot reservoir (heat source), in kelvin
- Tc — Absolute temperature of the cold reservoir (heat sink), in kelvin
Temperatures must be absolute (kelvin), never Celsius or Fahrenheit. The result is a fraction between 0 and 1; multiply by 100 for a percentage. Th must be greater than Tc and greater than 0.
Kelvin conversion
T(K) = T(°C) + 273.15
Convert Celsius to kelvin before using the efficiency formula. For Fahrenheit, first convert to Celsius with (°F − 32) × 5/9, then add 273.15.
Examples
- Input
- Th = 600 K, Tc = 300 K
- Result
- η = 50%
- Why
- η = 1 − Tc/Th = 1 − 300/600 = 1 − 0.5 = 0.50, which is 50%. This is the maximum efficiency; a real engine between these reservoirs would do worse due to friction and irreversibility.
- Input
- Th = 800 K, Tc = 300 K
- Result
- η = 62.5%
- Why
- η = 1 − 300/800 = 1 − 0.375 = 0.625, or 62.5%. Raising the hot-side temperature while keeping the cold side fixed increases the Carnot limit, which is why turbines run as hot as materials allow.
- Input
- Th = 373.15 K (100 °C), Tc = 298.15 K (25 °C)
- Result
- η ≈ 20.1%
- Why
- Convert first: 100 °C = 373.15 K, 25 °C = 298.15 K. Then η = 1 − 298.15/373.15 = 1 − 0.79901 = 0.20099 ≈ 20.1%. The small temperature gap explains why low-pressure steam engines are inefficient.
When to use this tool
- Estimating the best-case efficiency of a power plant, internal combustion engine, or turbine from its operating temperatures.
- Checking a homework or exam problem on thermodynamics and the second law where Carnot efficiency is required.
- Comparing how much efficiency you could gain by raising the hot-side temperature or lowering the cold-side temperature.
- Sanity-checking a manufacturer's efficiency claim — if a quoted figure beats the Carnot limit for its temperatures, the claim is wrong.
Common mistakes
- Using Celsius or Fahrenheit directly. The formula only works with absolute (kelvin) temperatures — always add 273.15 to a Celsius value first, or the answer is meaningless and can even go negative.
- Swapping the reservoirs so that Tc > Th. The hot reservoir Th must be the larger temperature; if Tc exceeds Th the formula gives a negative efficiency, which is physically impossible.
- Believing the Carnot value is achievable in practice. It is a theoretical upper bound for a reversible cycle; real engines lose energy to friction, turbulence, and heat leakage and reach only a fraction of it.
- Forgetting that efficiency is dimensionless. η is a ratio of work out to heat in, so the answer is a pure number (or percent), not joules or watts.
Frequently asked questions
Why must temperatures be in kelvin and not Celsius?
The Carnot formula is a ratio of absolute temperatures, which are measured from absolute zero. Celsius and Fahrenheit have arbitrary zero points, so dividing two Celsius values gives a nonsensical ratio. Convert with T(K) = T(°C) + 273.15 before calculating.
Can a real heat engine reach Carnot efficiency?
No. Carnot efficiency is the maximum possible for a perfectly reversible engine with no friction, no heat leakage, and infinitely slow operation. Every real engine has irreversibilities, so its actual efficiency is always lower — typical car engines reach roughly 25–35% even though their Carnot limit is much higher.
How do I improve a heat engine's efficiency?
Since η = 1 − Tc/Th, you raise efficiency by increasing the hot reservoir temperature Th or decreasing the cold reservoir temperature Tc. Raising Th is usually most effective, which is why modern turbines run at extreme temperatures, limited mainly by material strength.
What does an efficiency of 1 (100%) require?
η = 1 only if Tc = 0 K (absolute zero) or Th is infinite. Both are unreachable, so 100% efficiency is impossible. This is a direct consequence of the second law of thermodynamics: some heat must always be rejected to the cold reservoir.
Does the working substance (steam, air, etc.) change the Carnot efficiency?
No. A remarkable result of Carnot's theorem is that the maximum efficiency depends only on the two reservoir temperatures, not on the working fluid or engine design. Steam, gas, or any other substance running reversibly between the same two temperatures has the identical Carnot limit.
What is the difference between Carnot efficiency and thermal efficiency?
Thermal efficiency is the actual ratio of useful work output to heat input for a specific engine. Carnot efficiency is the theoretical ceiling set only by the reservoir temperatures. Dividing the real thermal efficiency by the Carnot efficiency gives the engine's 'second-law' or relative efficiency.
Sources & references
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